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Thermodynamic Formalism for Transient Potential Functions

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Abstract

We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and positive Radon eigenmeasures in forms of Martin kernels. These eigenmeasures can be characterized in terms of the direction of escape to infinity of their orbits, when viewed inside a suitable Martin-like compactification of the underlying shift space. We relate these results to first-order phase transitions in one-dimensional lattice gas models with infinite set of states. This work complements earlier works by Sarig (Ergod Theory Dyn Syst 19(06):1565–1593, 1999; Isr J Math 121(1):285–311, 2001) who focused on the recurrent scenario.

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Acknowledgements

I would like to express my gratitude to my adviser Prof. Omri Sarig for the professional and moral support. I would like also to thank the referees for the useful comments. This work is part of the author’s Ph.D. dissertation at the Weizmann Institute of Science and was also partially supported by Israel Science Foundation Grants 199/14 and 1149/18.

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Correspondence to Ofer Shwartz.

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Communicated by C. Liverani

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Shwartz, O. Thermodynamic Formalism for Transient Potential Functions. Commun. Math. Phys. 366, 737–779 (2019). https://doi.org/10.1007/s00220-019-03316-8

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